论文标题
五维para-cr歧管和接触型几何形状三维的几何形状
Five-dimensional para-CR manifolds and contact projective geometry in dimension three
论文作者
论文摘要
我们研究了$ 5 $维的para-cr结构的不变属性，其Levi形式的形式精确地是一个方向，并且是$ 2 $ nondementer的。我们意识到，此类结构的三个主要（基本）para-cr不变性中有两个是Monge（1810）和Wuenschmann（1905）\ [M（G）已知的经典差异不变性。 40G_{ppp}^3-45G_{pp}G_{ppp}G_{pppp}+9G_{pp}^2G_{ppppp}, \quad W(H) := 9D^2H_r-27DH_p-18H_rDH_r+18H_pH_r+4H_r^3+54H_z. \ \]消失的$ m（g）\ equiv 0 $为$（p，g）$ - 在圆锥中包含的函数的图形提供了局部必要条件，而消失的$ W（H）\ equiv 0 $为定义自然Lorentzian几分法的3rd order ode ode ode ode ode od ode od od od ode od ode od ode od ode od ode od of in in of and of-infly-if条件。 主要是，我们对我们类别的Para-CR结构的第三个基本不变性（最简单的阶段，最低顺序）和混合自然$ n（g，h）：= 2G_ {ppp}+g_ {pp} h_ {rr} $进行了几何解释。我们确定消失的$ n（g，h）\ equiv 0 $给出了para-cr歧管的两个$ 3 $维二维的标准的条件，其两个规范可集成的排名-2 $ 2 $分布，配备了接触式的几乎值。还讨论了我们在Arxiv：2003.08166中首先注意到Wuenschmann不变式与Monge“不变”之间的奇怪转变，并进一步揭示了其奥秘。
We study invariant properties of $5$-dimensional para-CR structures whose Levi form is degenerate in precisely one direction and which are $2$-nondegenerate. We realize that two, out of three, primary (basic) para-CR invariants of such structures are the classical differential invariants known to Monge (1810) and to Wuenschmann (1905) \[ M(G) := 40G_{ppp}^3-45G_{pp}G_{ppp}G_{pppp}+9G_{pp}^2G_{ppppp}, \quad W(H) := 9D^2H_r-27DH_p-18H_rDH_r+18H_pH_r+4H_r^3+54H_z. \] The vanishing $M(G) \equiv 0$ provides a local necessary and sufficient condition for the graph of a function in the $(p,G)$-plane to be contained in a conic, while the vanishing $W(H) \equiv 0$ gives an if-and-only-if condition for a 3rd order ODE to define a natural Lorentzian geometry on the space of its solutions. Mainly, we give a geometric interpretation of the third basic invariant of our class of para-CR structures, the simplest one, of lowest order, and of mixed nature $N(G,H):=2G_{ppp}+G_{pp}H_{rr}$. We establish that the vanishing $N(G,H) \equiv 0$ gives an if-and-only-if condition for the two $3$-dimensional quotients of the para-CR manifold by its two canonical integrable rank-$2$ distributions, to be equipped with contact projective geometries. A curious transformation between the Wuenschmann invariant and the Monge invariant, first noted by us in arXiv:2003.08166, is also discussed, and its mysteries are further revealed.
